Energy Density in Magnetic Field

The energy density in a magnetic field represents the amount of energy stored per unit volume in the magnetic field. It is denoted by the symbol U and is given by the formula:

U = (1/2)B⋅H

where B is the flux density and H is the magnetic field intensity.

Explanation

To understand why the energy density in the magnetic field is given by U = (1/2)B⋅H, let's consider a magnetic circuit with a magnetic field passing through a cross-sectional area A.

Magnetic Flux Density (B)
The magnetic flux density, B, represents the amount of magnetic flux per unit area. It is defined as the magnetic flux (Φ) passing through a surface divided by the cross-sectional area (A):

B = Φ/A

Magnetic Field Intensity (H)
The magnetic field intensity, H, represents the magnetomotive force per unit length. It is defined as the magnetomotive force (F) divided by the length (l) of the magnetic path:

H = F/l

Energy Density (U)
The energy density, U, represents the amount of energy stored per unit volume in the magnetic field. It is given by the formula:

U = (1/2)B⋅H

Derivation

To derive the formula for energy density, we consider a small volume element within the magnetic field. The volume element has a cross-sectional area A and a length Δl.

The energy stored in this volume element is given by:

ΔU = (1/2)B⋅H⋅A⋅Δl

The total energy stored in the entire magnetic field can be obtained by integrating this expression over the entire volume:

U = ∫(1/2)B⋅H⋅dV

Since B and H are constant within the magnetic field, we can take them out of the integral:

U = (1/2)B⋅H⋅∫dV

The integral of dV represents the total volume V of the magnetic field:

U = (1/2)B⋅H⋅V

Since V = A⋅l, where A is the cross-sectional area and l is the length of the magnetic path, we can substitute V with A⋅l:

U = (1/2)B⋅H⋅A⋅l

Therefore, the energy density in the magnetic field is given by U = (1/2)B⋅H.

Answer
The correct option is D) BH/2.